Hough transform neural network for pattern detection and seismic applications

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Hough transform neural network for pattern detection and

seismic applications

Kou-Yuan Huang

, Kai-Ju Chen, Jiun-Der You, An-Ching Tung

Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan

a r t i c l e

i n f o

Available online 25 June 2008 Keywords:

Hough transform neural network Seismic pattern detection Reﬂection wave

Hyperbolic pattern detection

a b s t r a c t

Hough transform neural network is adopted to detect the line pattern of direct wave and the hyperbolic pattern of reﬂection wave in a one-shot seismogram. We use time difference from point to hyperbola and line as the distance in the pattern detection of seismic direct and reﬂection waves. This distance calculation makes the parameter learning feasible. One set of parameters represents one pattern. Many sets of parameters represent many patterns. The neural network can calculate the distances from point to many patterns as total error. The parameter learning rule is derived by gradient descent method to minimize the total error. The network is applied to three kinds of data in the experiments. One is the line and hyperbolic pattern in the image data. The second is the simulated one-shot seismic data. And the last is the real one-shot seismic data. Experimental results show that lines and hyperbolas can be detected correctly in three kinds of data. The method can also tolerate certain level of noise data. The detection results in the one-shot seismogram can improve the seismic interpretation and further seismic data processing.

1. Introduction

Hough transform (HT) was used to detect the parameterized shapes by mapping original image data in the image space into the parameter space[5,6,10,12]. The purpose of HT was to ﬁnd the peak value (maximum) in the parameter space. The coordinates of a peak value in parameter space were corresponding to a shape in the image space. Calculating such transformation was time consuming, the memory in parameter space was too large, and the peak determination was not easy in the parameter space.

Neural network was developed to solve the HT problem[1–3]. The Hough transform neural network (HTNN) was designed for detecting lines, circles, and ellipses[1–3]. The determination of parameters of objects was by the neural network learning, not by the mapping to the parameter space. But there was no application to the detection of hyperbola.

Seismic pattern recognition plays an important role in oil exploration. In a one-shot seismogram, the travel-time curve of direct wave pattern is a straight line and the reﬂection wave pattern is a hyperbola. In 1985 and 1987, Huang et al. had applied the HT to detect the line pattern of direct wave and the hyperbolic pattern of reﬂection wave in a one-shot seismogram [7,8]. However, it had the same serious drawbacks in the calculation as the conventional HT.

In 2006, Huang et al. [9] applied HTNN to detect the line pattern of direct wave and the hyperbolic pattern of reflection wave in a one-shot seismogram. However, the distance from point to line and hyperbola using different distance definitions, determination of two-stage learning and decreasing basis in Gaussian basis function make the learning complex. Here, we also take the advantage of HTNN. We define the vertical time difference as the distance from point to line and to hyperbola that makes the definition consistence. Moreover, the learning process is simplified to one-stage and the preprocessing of the envelope processing, threshold processing, and peak detection processing is taken to improve the detection result. We also have the real seismic data experiment.

2. Seismic signals

Seismic exploration is an essential procedure in oil and gas exploration. Seismic signal is produced by explosion and receiv-ing. After explosion, there are direct waves along the ground surface and reﬂection waves from the reﬂection layer.

2.1. Direct wave

There are two basic kinds of seismic waves: the direct wave and the reﬂection wave[4,13,14]. Direct wave propagates along the surface to the receivers.Fig. 1shows the wave path of the Contents lists available atScienceDirect

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direct wave, and the time when the direct wave reaches the receiver is in Eq. (1), where x is the distance from the source to the receiver.Fig. 2is the direct wave in the one-shot seismogram with 40 traces. The time–distance curve is a line. Wave velocity in the layer is 2200 m/s, and the distance of the receiving station is 50 m. The sampling interval is 0.004 s:

t ¼x

v. (1)

2.2. Reﬂection wave

The reflection wave propagates through the medium and is reflected back from the reflection layer. There are two cases: from the horizontal reflection layer and from the dipping reflection layer. The first case is the horizontal reflection layer as shown in

Fig. 3. The time of the wave reaching the kth receiver is derived in Eq. (2), and the time–distance curve is a hyperbola as shown in

Fig. 4: t ¼ðOQ þ QPÞ v ¼ O0_P v ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi OO0 2 þOP2 q v ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2dÞ2 þx2 q v . (2)

The other case is a dipping reflection layer as shown inFig. 5. In this case, we use trigonometric laws of cosine to derive the relation between receiving time t and the distance x in Eq. (3). After further algebraic calculation, we find it is also a hyperbola (Eq. (4)).Fig. 6shows the one-shot seismogram of the reflection wave from the dipping reflection layer. Note here, the wave from dipping reflection layer shifts right related to the wave which is

from horizontal reﬂection layer:

t ¼ ðOQ þ QPÞ v ¼ O0_P v ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi O0 O2þOP22O0 OOP cosðangleðffO0 OPÞÞ q v ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi O0 O2þOP22O0 OOP cosð90

y

Þ q v ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2dÞ2þx2_{4dx sin}

_y

q v , (3) 1 2 n Source Receiver line Ground Under the Earth k

Direct wave

| x |

v Fig. 1. Wave path of the direct wave.

0 5 10 15 20 25 30 35 40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Station Time (Sec.)

Fig. 2. Direct wave in the one-shot seismogram.

1 2 n Receiver Ground Horizontal reflecting layer k O O’ W Q P d d | x | Source

Fig. 3. Reﬂection from the horizontal reﬂection layer.

0 5 10 15 20 25 30 35 40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Station Time (Sec.)

Fig. 4. One-shot seismogram: reﬂection wave from the horizontal reﬂection layer.

n Source

Receiver

Dipping reflecting layer O Ground h d 2 1 k T W Q P d O’ x v

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t ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2dÞ2_{þ ðx 2d sin}

_y

Þ2 ð2dÞ2_sin2

y

q v ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2d cos

y

Þ2þ ðx 2d sin

y

Þ2 q v . (4)

3. Seismic pattern detection system

To detect parameters of the line (direct wave) and hyperbola (reﬂection wave), we adopt the HTNN. The proposed seismic pattern detection system is shown inFig. 7.

3.1. Preprocessing

The input seismogram including the direct wave and reﬂection wave in Fig. 8 passes through envelope processing, threshold processing, and peak detection processing[7]. For seismic data, s(xi, ti), 1pxip65, 1ptip512, after envelope processing, s(xi, ti)

becomes s0_(x

i, ti), we set a threshold T. Through threshold

processing, s0_(x

i, ti)XT, and peak detection processing, data

become the object points, xi¼[xi, ti]T, i ¼ 1, 2, y, n.Fig. 9is the

result of preprocessing. After preprocessing, the object points enter the HTNN.

3.2. Hough transform neural network (HTNN)

The adopted HTNN consists of three layers: distance layer, radial basis function (RBF) layer, and the total error layer. The network is shown inFig. 10. It is an unsupervised network capable of detecting m parameterized objects: lines and hyperbolas, simultaneously. The objects can ﬁt the data. And the error must be minimized. Gradient descent method is used in the parameter learning.

InFig. 10, input vector xi¼[xi, ti]Tis the ith point of the image,

where i ¼ 1, 2, y, n. In the preprocessed seismic image, xiis the

trace index between shot point and receiving station, and ti is

index in time coordinate. Each pattern (line or hyperbola) has one set of parameters. The number of patterns, m, must be given before HTNN learning. Input each point xiinto distance layer, we

calculate the distance dik¼Dk(xi) ¼ Dk(xi, ti) from xi to the kth

pattern (line or hyperbola), k ¼ 1, 2, y, OLor OH, where OLis the

number of line patterns and OH is the number of hyperbolic

patterns. Then, dikpasses through the RBF layer and the activation

output is eik¼1f(dik), where f( ) is a Gaussian basis function, i.e.,

f ðdikÞ ¼expðd2ik2=

s

2Þ, and eikis the error or the modiﬁed distance

from the ith point xito the kth pattern. Thus, when dikis near zero,

eikis also near zero. In the last layer, we calculate the total error

from xito all m patterns, Ei¼ei1ei2 ? eim. When xibelongs to

one pattern, then eik¼0, and Ei¼0.

3.3. Distance layer

3.3.1. Distance from point to hyperbola

In a one-shot seismogram, the reﬂection wave pattern is a hyperbola. Note that, for seismic pattern, there is no rotated hyperbola. So the equation of the kth hyperbola is

x x0;k ak 2 þ t t0;k bk 2 1 ¼ 0. (5)

We consider the positive x side, the equation becomes:

HkðxÞ ¼ bk ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x x0;k ak 2 þ1 s ðt t0;kÞ ¼0. (6)

We minimize the distance from point xito the hyperbola. That is

minimize1 2jjx xijj

2_{subject to Hk}_{ðxÞ ¼ 0.} ₍₇₎ From Lagrange method, the Lagrange function is

lðx;

l

Þ ¼1 2jjx xijj 2 þ

l

bk ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x x0;k ak 2 þ1 s ðt t0;kÞ 2 4 3 5. (8) By the first-order necessary condition, we have the following equations:

q

xlðx;

l

Þ ¼ ðx xiÞ þ

l

bk a2 k ðx x0;kÞ , ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx x0;k=akÞ2þ1 q ¼0,

q

tlðx;

l

Þ ¼ ðt tiÞ

l

¼0,

q

ql

lðx;

l

Þ ¼bk ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x x0;k ak 2 þ1 s ðt t0;kÞ ¼0. (9)

To solve the equations in Eq. (9), we must take fourth-order complex computations. So alternatively we consider the time difference as the distance from a point xi¼(xi, ti) to a hyperbola in

seismic pattern detection case. The distance is deﬁned as

dik¼tðxiÞ ti¼bk ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xix0;k ak 2 þ1 s ðtit0;kÞ ¼HkðxiÞ. (10)

Fig. 11 illustrates the distance from point to hyperbola in the difference of vertical time direction.

0 5 10 15 20 25 30 35 40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Station Time (Sec.)

Fig. 6. One-shot seismogram: reﬂection wave from the dipping reﬂection layer.

Preprocessing Hough transform

neural network Detected parameters Detected patterns Input one-shot seismogram

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3.3.2. Distance from point to line

Although Basak and Das proposed HTNN to detect lines, they used the second-order equation of conoidal shapes[2]. Here, we use the slope–intercept form of straight-line equation in the analysis.

The equation of the kth line is

t ¼ mkx þ bk, (11) or in the form:

LkðxÞ ¼ mkx t þ bk¼0. (12) The distance from the point xi to the kth line is a constraint

minimization problem:

minimize1 2jjx xijj

2_{subject to Lk}_{ðxÞ ¼ 0.} ₍₁₃₎ However, for the case of seismic pattern, to have the same distance measure, the distance from a point to the line is also considered as time difference from the point xi¼(xi, ti) to the line.

The distance used here is

dik¼tðxiÞ ti¼mkx tiþbk¼LkðxiÞ. (14)

Fig. 12illustrates the distance from point to line in the difference of vertical time direction.

Fig. 8. One-shot seismogram.

Fig. 9. Result of preprocessing.

xi ti Ei di1 dik dim … … … … … … ei1 … eik … eim Distance Layer : distance from point to each pattern RBF Function Layer

Total Error Layer

Δp 1 b -1 m Ei = ei1 · ei2 · … · eik · … · eim

Fig. 10. Hough transform neural network.

(x_i, t_i) + 1 + t0, k ak x − x0, k t(x) = bk Hk (x) = 0 2 (xi, t (xi)) t (xi) − ti= bk = Hk (xi) + 1 − (ti− t0, k) ak x − x0, k 2

Fig. 11. Distance from point to hyperbola.

(xi, ti) t (x) = mx + b Lk (x) = 0 (xi, t (xi)) t (xi) − ti= mx + b − ti = Lk (xi) Fig. 12. Distance from point to line.

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3.4. RBF function layer

After the distance calculation from a point to a pattern, the modiﬁed distance or the error is deﬁned as

eik¼1 f ðdikÞ ¼1 exp d 2 ik

s

2 !

, (15)

where f(dik) is the RBF. The

s

in Gaussian basis function controls

the effected range. For the larger

s

, the effected range is larger. If we do not use RBF, we can use f(dik) ¼ |dik|.

3.5. Total error layer

We consider that the multiplication of m errors from RBF function layer is the total error. So the total error for point xiis

deﬁned as

Ei¼Cðei1; . . . ; eik; . . . ; eimÞ ¼ei1ei2 eim¼ Y 1pkpm

eik. (16)

The total error becomes zero when the distance between point xi

and any pattern is zero, i.e., eik¼0, and Ei¼0.

4. Parameter learning rules

Initially give random values to parameters of each pattern (line or hyperbola). For each input xi, we calculate the distance from xi

to each pattern. Then calculate the total error, Ei, of xi, and use

gradient descent method to update parameters of each pattern to reach minimum error.Fig. 13illustrates the change of parameters corresponding to the change of the pattern, i.e., pattern matches the data by learning. The parameters of line or hyperbola can be written as a parameter vector pk, and the learning

pk(t+1) ¼ pk(t)+

D

pk(t), where (k ¼ 1, 2, y, m), and by gradient

descent method:

D

pk¼

b

q

Ei

q

pk

, (17)

where

b

is the learning rate. From Eq. (17) and by chain rules,

D

pk

can be written as

D

pk¼

b

q

Ei

q

eik

q

eik

q

dik

q

dik

q

pk ¼

b

Ei eik 2dik

s

2 ð1 eikÞ

q

dik

q

pk . (18)

We derive qdik/qpkfor hyperbola and line, respectively, as follows.

4.1. Learning rule for hyperbola

For the non-rotated kth hyperbola, or seismic reﬂection pattern, in Eq. (6), the parameter vector of hyperbola is pk¼ ½ ak bk x0;k t0;kT, and thus

q

dik

q

pk ¼

q

dik

q

ak

q

dik

q

bk

q

dik

q

xk

q

dik

q

tk " # . (19)

From Eq. (10), the elements of qdik/qpkare

q

dik

q

ak¼

ððbk=akÞÞððxix0;kÞ=akÞ2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ððxix0;k=akÞÞ2þ1

q ,

Input one point:

Pattern of p = (m,b) Pattern of p’ = (m’, b’)

Input the next point:

Pattern of p’= (m’, b’) Pattern of p’’= (m’’, b’’)

p' = p + Δp

p'' = p' + Δp

Fig. 13. Illustration of parameter learning.

For i = 1, 2, …, n Start

No

1. Input xi to network

2. Calculate distance (dik) from xi to each object (line and hyperbola)

3. Calculate RBF function output f(dik) 4. Calculate total error Ei

1. Calculate

∂pk ∂Ei Δpk = − 2. Adjust parameter vector

pk (t + 1) = pk (t) + Δpk (t) Ei

1. Set OL and OH, the number of line and hyperbola.

2. Initialize random parameter vectors.

Preprocessing (Thresholding) {x1, x2, …, xn} s (x, t) {E1, E2, …, En} Yes End ? 1 ∑n i = 1 Ei< Eth n Eavg=

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q

dik

q

bk¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xix0;k ak 2 þ1 s ,

q

dik

q

x0;k

¼ððbk=akffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÞÞððxix0;kÞ=akÞ ððxix0;k=akÞÞ2þ1

q ,

q

dik

q

t0;k¼1. (20)

Then, from Eqs. (18) and (20), we have

D

pk¼ ½

D

ak

D

bk

D

x0;k

D

t0;kT ¼

b

Ei eik _2d ik

s

2 ð1 eikÞ ððbk=akÞÞððxix0;kÞ=akÞ2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ððxix0;kÞ=akÞ2þ1 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xix0;k ak 2 þ1 r ððbk=akÞÞððxix0;kÞ=akÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ððxix0;kÞ=akÞ2þ1 q 1 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 . (21)

For seismic reﬂection wave from the horizontal reﬂection layer, we have x0,k¼0 in Eq. (6). So the parameter vector is

pk¼ ½ ak bk t0;kT, and by Eq. (21), which implies parameter

adjustment:

D

pk¼ ½

D

ak

D

bk

D

t0;kT ¼

b

Ei eik 2dik

s

2 ð1 eikÞ

ððbk=akÞÞððxix0;kÞ=akÞ2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ððxix0;kÞ=akÞ2þ1 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xix0;k ak 2 þ1 r 1 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 . (22)

4.2. Learning rule for line

For the kth line, the parameter vector is pk¼ ½ mk bkT. Thus,

q

dik

q

pk ¼

q

dik

q

mk

q

dik

q

bk " #T . (23)

From Eq. (14), we can get:

q

dik

q

mk¼xi and

q

dik

q

bk¼1. (24)

Hence, from Eqs. (18) and (24), we have

D

pk¼

D

mk

D

bk h iT ¼

b

Ei eik _2dik

s

2 ð1 f ðdikÞÞ xi 1 T . (25)

Fig. 15. Experiments on lines: (a) four lines, (b) data with Gaussian noise N(0, 1), (c) data with Gaussian noise N(0, 2), and (d), (e), (f) corresponding error plot with iterations.

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4.3. Learning process

The ﬂowchart of the learning system is shown inFig. 14. Here, we use simple one-stage learning instead of two-stage learning

[1], that is, all parameters are changed simultaneously. The number of patterns or objects, including lines and hyperbolas, OLand OH, must be given. Initially give random parameter vectors.

Then input each data and adjust the parameter vector as in Eq. (21) or (22) and (25). One input data has one error. Then we calculate the average error for all input data. Finally, if the average error is less than a threshold, Eth, or the iterations reach the preset

maximum iteration number, the learning stops.

5. Experiments

In the first part, several lines and hyperbolas are generated and detected. In the second part, we do the experiments on the simulated one-shot seismogram for the line detection of direct wave and the hyperbola detection of reflection wave. In the third part, we do the experiments on the detection of direct wave and the reflection wave in the real seismic data.

5.1. Experiments on the detection of line and hyperbolic patterns in simulated image data

We do the experiments on the detection of lines, hyperbolas, and both of them inFigs. 15–17. The learning rates of the center (x0, y0), the major and minor axes (a and b) of hyperbola, the slope

(m) and intercept (b) of line are not the same and set by experiments. In experiments of part one, learning rates for the center is 1, for the major and minor axes of hyperbola is 0.5, for the slope of line is 0.01 and for the intercept of line is 1. The

s

in Gaussian basis function is preset and is found to be related to the size of images. In experiment part one,

s

is 10.

InFig. 15(a), there are four lines, OL¼4, with total 200 points.

In Fig. 15(b), the data are disturbed by Gaussian random noise with zero mean and variance one, that is, N(0,1). InFig. 15(c), the Gaussian noise is N(0, 2). We see that lines inFig. 15(a) and (b) are well detected even ifFig. 15(b) has noise data. But inFig. 15(c) larger noise data, the detection result is affected by noise.

InFig. 16(a), there are two hyperbolas (100 points), OH¼2. And

inFig. 16(b) and (c), data are with Gaussian random noise N(0,1), and N(0, 2), respectively. Results show that hyperbolas are detected correctly inFig. 16(a)–(c).

In Fig. 17(a), there are two lines, OL¼2, and two hyperbolas,

OH¼2, with total 200 points, and data with Gaussian random

noise N(0, 1) and N(0, 2) inFig. 17(b) and (c). Results show that hyperbolas are detected correctly inFig. 17(a)–(c).

The parts (d), (e), and (f) ofFigs. 15–17are plots of error versus iterations for parts (a), (b), and (c) ofFigs. 15–17, respectively.

5.2. Experiments on simulated seismic data

5.2.1. Horizontal reﬂection layer

The HTNN is applied to the simulated seismic data for line and hyperbolic detection.Fig. 8is the simulated one-shot seismogram from the horizontal reﬂection layer where the depth of the

Fig. 16. Experiments on lines: (a) two hyperbolas, (b) data with Gaussian noise N(0,1), (c) data with Gaussian noise N(0, 2), and (d), (e), (f) corresponding error plot with iterations.

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reflection layer is 500 m and the velocity of the p-wave in the sedimentary rock is about 2500 m/s[11]. There are 65 receiving stations on both sides of explosion with 50 m between each two receiving stations. The sampling interval is 0.004 s. The impulse response is 25 Hz Ricker wavelet. Reflection coefficient is 0.2 and noise is band-passed noise, 10.2539–59.5703 Hz, with uniform

distributed over (0.2, 0.2). The one-shot seismogram inFig. 8is ﬁrst preprocessed by envelope processing, threshold processing, and peak detection processing. The preprocessing result is shown inFig. 9. The image size is 512 65 where the origin is on the top-left corner with horizontal x-axis and vertical y-axis. These 129 points are then used as the input data to HTNN.

Fig. 18. (a) Line and hyperbola in Fig. 9 detected by HTNN. (b) Plot of error versus iterations.

Fig. 17. Experiments on lines and hyperbolas: (a) two lines and two hyperbolas, (b) data with Gaussian noise N(0,1), (c) data with Gaussian noise N(0, 2), and (d), (e), (f) corresponding error plot with iterations.

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In this part of the experiment, learning rates for the center is 0.1, for the major and minor axes of hyperbola is 0.1, for the slope of line is 0.1 and for the intercept of line is 5. The

s

in Gaussian basis function is 25. The preset numbers of lines and hyperbolas are OL¼2 and OH¼1.Fig. 18(a) shows the result of lines and the

hyperbola detection in Fig. 9. And the plot of error versus iterations is shown inFig. 18(b).

5.2.2. Dipping reﬂection layer

Fig. 19shows the detection of direct and reﬂection wave from the dipping reﬂection layer by HTNN.Fig. 19(a) is the original one-shot seismogram. After preprocessing, the input data inFig. 19(b) have 127 points. The detection results are inFig. 19(c). And the error plot is shown inFig. 19(d). The result is also good.

5.3. Experiments on real seismic data

The system is also applied to detect direct wave and reﬂection wave in real seismic data. We obtain data from Seismic Unix System developed by Colorado School of Mines[14].

The real data with the size 3100 48 shown in Fig. 20(a) is from Canadian Arctic, which has 48 traces and 3100 samples per trace with sampling interval 0.002 s. The horizontal axis is the trace number and the vertical axis is time t. After preprocessing

[7],Fig. 20(b) shows the result of preprocessing with the threshold 0.15. We only choose 53 points with yo500 which includes points

from direct wave and ﬁrst reﬂection wave as input data to the HTNN. Learning rates for the center is 0.1, for the major and minor axes of hyperbola is 10, for the slope of line is 1, and for the intercept of line is 100. The

s

in Gaussian basis function is 80. The preset numbers of lines and hyperbolas are OL¼2 and OH¼1. Fig. 20(c) is the detection result of two lines and a hyperbola. The plot of the result on the original one-shot seismogram is shown in

Fig. 20(d). The detected parameters of lines of direct wave and hyperbola of reﬂection wave in image space inFig. 20(c) are listed inTable 1.

6. Conclusions

HTNN is adopted to detect the line pattern and the hyperbola pattern in image data, and is also adopted to detect the line pattern of direct wave and the hyperbola pattern of reflection wave in a one-shot seismogram. The objects can fit the data. The parameter learning rules are derived by gradient descent method to minimize the total error. We define the vertical time difference as the distance from point to hyperbola that makes the learning feasible. In the experiments on the pattern detection of the line and hyperbola, patterns are well-detected even if there are noises. The method can tolerate certain level of noise. For the data with large noise, the method may suffer from the local optimal problem. In the experiments on seismic data, the detection results in the line pattern of direct wave and the hyperbola pattern of

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reﬂection wave are good. The detection results can improve seismic interpretation and further seismic data processing.

In the experiments, the detection results depend on the learning rate and parameter

s

in the Gaussian basis function. These parameters are set heuristically. Further research may be done to justify why these selected values provide good results in the experiments.

Acknowledgments

The authors would like to thank the Colorado School of Mines for the use of Seismic Unix System and the real data. This work

was supported in part by the National Science Council, Taiwan, under NSC-94-2213-E-009-133 and NSC-95-2221-E-009-221.

References

[1] J. Basak, A. Das, Hough transform networks: learning conoidal structures in a connectionist framework, IEEE Trans. Neural Netw. 13 (2) (2002) 381–392. [2] J. Basak, A. Das, Hough transform network: a class of networks for identifying

parametric structures, Neurocomputing 51 (2003) 125–145.

[3] G.L. Dempsey, E.S. McVey, A Hough transform system based on neural networks, IEEE Trans. Ind. Electron. 39 (1992) 522–528.

[4] M.B. Dobrin, Introduction to Geophysical Prospecting, third ed., McGraw-Hill, 1976.

[5] R.O. Duda, P.E. Hart, Use of the Hough transform to detect lines and curves in pictures, Commun. Assoc. Comput. Mach. 15 (1972) 11–15.

[6] P.V.C. Hough, Method and Means for Recognizing Complex Patterns, US Patent 3069654, 1962.

[7] K.Y. Huang, K.S. Fu, T.H. Sheen, S.W. Cheng, Image processing of seismo-grams: (A) Hough transformation for the detection of seismic patterns. (B) Thinning processing in the seismogram, Pattern Recognit. 18 (6) (1985) 429–440.

[8] K.Y. Huang, T.H. Sheen, S.W. Cheng, Z.S. Lin, K.S. Fu, Seismic image processing: (I) Hough transformation, (II) thinning processing, (III) linking processing, handbook of geophysical exploration: section I, Seism. Explor., Pattern Recognit. Image Process. 20 (1987) 79–109.

[9] K.Y. Huang, J.D. You, K.J. Chen, H.L. Lai, A.J. Don, Hough transform neural network for seismic pattern detection, in: International Joint Conference on Neural Networks, July 16–21, Vancouver, Canada, 2006, pp. 4670–4675. [10] J. Illingworth, J. Kittler, Survey: a survey of the Hough transform, Comput. Vis.,

Graph., Image Process. 44 (1988) 87–116.

Fig. 20. (a) One-shot seismogram from Canadian Artic. (b) Result of envelope threshold and peak detection process. (c) Detection result of peaks with yo500. (d) Detected lines and the hyperbola superimposed on the one-shot seismogram.

Table 1

Detected parameters in Fig. 20(c) in image space 3100 48

Hyperbola of reﬂection wave x0 y0 a b

24.69 23.19 10.65 240.96

Lines of direct wave m b

24.28 583.48

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[11] P. Kearey, M. Brooks, An Introduction to Geophysical Exploration, Blackwell Science Publications, Oxford, 1991.

[12] V.F. Leavers, Survey: which Hough transform, Comput. Vis., Graph., Image Process. 58 (2) (1993) 250–264.

[13] M.M. Slotnick, Seismic Computing, The Society of Exploration Geophysicists, 1959.

[14] O. Yilmaz, Seismic Data Processing, The Society of Exploration Geophysicists, Tulsa, 1987.

Kou-Yuan Huang received the B.S. in physics (1973) and M.S. in geophysics (1977) from the National Central University, Taiwan, and M.S.E.E. (1980) and Ph.D. in electrical and computer engineering (1983) from Purdue University. From September 1983 to August 1988, he joined the faculty in the Department of Computer Science, University of Houston-University Park. Now he is the Professor at the Department of Computer Science at National Chiao Tung University, Hsinchu, Taiwan. He widely published papers in journals: Geophysics, Geoexploration, Pattern Recogni-tion, IEEE Transactions on Geoscience and Remote Sensing, Computer Processing of Oriental Languages, etc. He has authored two books, Neural Networks and Pattern Recognition, Weikeg Publishing Co., Taipei, Taiwan, 2000, 2nd ed., 2003, and Syntactic Pattern Recognition for Seismic Oil Exploration, World Scientiﬁc Publishing Co., Singapore, in Series in Machine Perceptron and Artiﬁcial Intelligence, Vol. 46, 2002. His major

contributions are in the areas of seismic pattern recognition using signal and image processing, statistical, syntactic, neural networks, fuzzy logic, and genetic methods.

Kai-Ju Chen received the B.S. in computer science (2005) and M.S. in multimedia engineering (2007) from the National Chiao Tung University, Taiwan. His research interests are neural network, pattern recogni-tion, and signal processing.

Jiun-Der You received the M.S. in computer science (2004) from the National Chiao Tung University, Taiwan.

An-Ching Tung received the B.S. in computer science (2005) from National Central University, Taiwan, and M.S. in multimedia engineering (2007) from the National Chiao Tung University, Taiwan.